A manifold Fueter-Sce phenomenon in one hypercomplex variable

TL;DR

This paper extends the Fueter-Sce phenomenon to a broader class of strongly T-regular functions over general associative *-algebras, revealing a new multi-step process and a novel one-variable hypercomplex phenomenon.

Contribution

It generalizes the Fueter-Sce phenomenon to strongly T-regular functions in associative *-algebras and uncovers a new hypercomplex phenomenon in one variable.

Findings

The Fueter-Sce phenomenon applies to strongly T-regular functions over associative *-algebras.

The symmetry involved is multi-axial, as introduced by Eelbode.

A new hypercomplex phenomenon in one variable is discovered.

Abstract

Fueter's theorem states, in modern terms, that the Laplacian maps slice-regular quaternionic functions into Fueter-regular functions with axial symmetry. This phenomenon is also present in the Clifford setting, where both slice-monogenic functions and generalized partial-slice monogenic are mapped by the Laplacian into monogenic functions with axial symmetry. These results are due, respectively, to Sce and Qian and to Xu and Sabadini. The present work puts the Fueter-Sce phenomenon into context for the wider class of strongly $T$-regular functions. It shows that the phenomenon appears over general associative $*$-algebras. Moreover, the symmetry considered here is multi-axial in a sense introduced by Eelbode. Additionally, but more surprisingly, the phenomenon studied by Fueter, Sce, Xu and Sabadini turns out to be the last step in a multi-step process. A new phenomenon in one hypercomplex variable is therefore discovered.